The lifespans of seals in a particular zoo are normally distributed. The average seal lives $17.4$ years; the standard deviation is $3.5$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living less than $10.4$ years.
$17.4$ $13.9$ $20.9$ $10.4$ $24.4$ $6.9$ $27.9$ $95\%$ $2.5\%$ $2.5\%$ We know the lifespans are normally distributed with an average lifespan of $17.4$ years. We know the standard deviation is $3.5$ years, so one standard deviation below the mean is $13.9$ years and one standard deviation above the mean is $20.9$ years. Two standard deviations below the mean is $10.4$ years and two standard deviations above the mean is $24.4$ years. Three standard deviations below the mean is $6.9$ years and three standard deviations above the mean is $27.9$ years. We are interested in the probability of a seal living less than $10.4$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $95\%$ of the seals will have lifespans within 2 standard deviations of the average lifespan. The remaining $5\%$ of the seals will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({2.5\%})$ will live less than $10.4$ years and the other half $({2.5\%})$ will live longer than $24.4$ years. The probability of a particular seal living less than $10.4$ years is ${2.5\%}$.